State Transition Matrix, Homogeneous and Forced Time Solutions

Lesson 2.1 — Finite-horizon quadratic objectives

Unit: Optimal control and quadratic design. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

Finite-horizon control chooses an input sequence that balances state deviation and control effort during a known time window. For a discrete linear model, the basic objective adds nonnegative state penalties and positive input penalties, plus a final-state penalty. The matrices Q, R, and Qf are engineering weights, not automatically physical energy. Their dimensions must be chosen so the sum is coherent. Feasible inputs and dynamics must also be respected.

Governing relation: J=sum_{k=0}^{N-1}(x_k^T Q x_k+u_k^T R u_k)+x_N^T Q_f x_N

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

For one-step scalar dynamics x1=x0+u0, x0=2, Qf=4 and R=1, minimize J=u0²+4(2+u0)². Differentiating gives 2u0+8(2+u0)=0, so u0=−1.6 and x1=0.4. The cost is 2.56+0.64=3.20. With u0=−2, the state reaches zero but the cost is 4; the optimum deliberately retains some error to use less input.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

An optimal input relative to one cost may be undesirable under another. Changing the scale used for x without adjusting Q changes the physical preference and the minimizer. The terminal penalty helps prevent a finite-horizon controller from ignoring end-of-horizon state.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

A quadratic penalty alone cannot enforce a strict input cap. If |u|≤1 in this problem, the unconstrained optimum is unavailable and the constrained optimum is u=−1, which yields cost 1+4=5.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Actuated environmental chamber

A model predicts temperature from heater power and measured wall temperature. A quadratic tracking penalty must be balanced against heater power and its rate of change, but power and temperature possess different units. Scale the weights and report a real input constraint. If the heater reaches its ceiling during warm-up, the unconstrained optimal controller is no longer the implemented law. An experiment can compare energy use, settling and temperature overshoot on the same synthetic disturbance trace.

Lab protocol: Cost accounting. Construct a table with physical units of each state and input; then show how the selected Q and R weights make their quadratic contributions comparable. Repeat at a second scale or command magnitude. If the optimum moves after simply changing reporting units, the weight transformation was inconsistent.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: Repeat the one-step problem for R=4 and Qf=4. Compute the optimum and interpret how the increased input penalty changes the residual state. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Bellman recursion and the Riccati step. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 2.2 — Bellman recursion and the Riccati step

Unit: Optimal control and quadratic design. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

Dynamic programming decomposes an optimal sequence into a current decision and the optimal remaining cost. For a linear quadratic problem, the remaining value function is quadratic in the state and its coefficient follows a Riccati recursion. The recursion emerges from completing the square; it is not merely a software command. At each time step the controller must respect the assumed dynamics and the positive-definite input penalty.

Governing relation: P_k=Q+A^T P_{k+1} A-A^T P_{k+1} B(R+B^T P_{k+1} B)^{-1}B^T P_{k+1} A

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

For scalar A=1, B=1, Q=1, R=1 and terminal P2=1, the last-step gain is K1=P2/(R+P2)=1/2 and P1=1+1−1/2=1.5. At the earlier step K0=P1/(1+P1)=1.5/2.5=0.6. The optimal two-step feedback changes with the remaining horizon: u0=−0.6x0 while u1=−0.5x1.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

The result illustrates a time-varying gain under a finite terminal date. It does not conflict with time-invariant infinite-horizon LQR; that is a different problem in which the Riccati recursion converges under suitable stabilizability and detectability assumptions.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

If R=0 and B^T P B is singular, the displayed inverse may not exist. Blindly calling matrix inversion without checking conditioning is a numerical and conceptual mistake.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Actuated environmental chamber

A model predicts temperature from heater power and measured wall temperature. A quadratic tracking penalty must be balanced against heater power and its rate of change, but power and temperature possess different units. Scale the weights and report a real input constraint. If the heater reaches its ceiling during warm-up, the unconstrained optimal controller is no longer the implemented law. An experiment can compare energy use, settling and temperature overshoot on the same synthetic disturbance trace.

Lab protocol: Cost accounting. Construct a table with physical units of each state and input; then show how the selected Q and R weights make their quadratic contributions comparable. Repeat at a second scale or command magnitude. If the optimum moves after simply changing reporting units, the weight transformation was inconsistent.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: Starting with P2=2 and Q=R=A=B=1, compute P1 and K1. Explain why the gain changes relative to the example. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Infinite-horizon LQR and a scalar analytic result. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 2.3 — Infinite-horizon LQR and a scalar analytic result

Unit: Optimal control and quadratic design. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

For a stabilizable state-space model and suitable nonnegative state weights, an infinite-horizon linear quadratic regulator minimizes the integral of state and input penalties. The continuous-time algebraic Riccati equation yields the gain u=−Kx. In a scalar stable or unstable system this can be solved analytically, revealing the tradeoff between performance and effort. The solution is model-dependent and assumes an unconstrained linear actuator.

Governing relation: A^T P+P A-P B R^{-1}B^T P+Q=0; K=R^{-1}B^T P

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

Let x_dot=−x+u, Q=1, R=1. The Riccati equation is −2P−P²+1=0, giving P=−1+√2≈0.4142 as the nonnegative stabilizing solution. Thus K≈0.4142 and the closed-loop pole A−BK≈−1.4142. The negative root P=−1−√2 is rejected because it is not the nonnegative stabilizing solution.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

The LQR gain can be interpreted by substituting it into the dynamics: state decays faster than the open-loop pole of −1 but requires input activity. If the model mistakenly says actuator gain B is twice its actual value, the predicted closed-loop speed will be wrong.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

A typical solver can return a matrix P without revealing whether the given pair is stabilizable. An uncontrollable unstable mode cannot be rescued by choosing enormous Q.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Actuated environmental chamber

A model predicts temperature from heater power and measured wall temperature. A quadratic tracking penalty must be balanced against heater power and its rate of change, but power and temperature possess different units. Scale the weights and report a real input constraint. If the heater reaches its ceiling during warm-up, the unconstrained optimal controller is no longer the implemented law. An experiment can compare energy use, settling and temperature overshoot on the same synthetic disturbance trace.

Lab protocol: Cost accounting. Construct a table with physical units of each state and input; then show how the selected Q and R weights make their quadratic contributions comparable. Repeat at a second scale or command magnitude. If the optimum moves after simply changing reporting units, the weight transformation was inconsistent.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: Solve the same scalar problem for Q=4 and R=1. Select the stabilizing Riccati root and calculate the new closed-loop pole. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Reference tracking, prefilters and control constraints. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Lesson 2.4 — Reference tracking, prefilters and control constraints

Unit: Optimal control and quadratic design. Prerequisites: Book 11 feedback and state-space fundamentals; preceding lessons in this unit. Learning objectives: explain the model, reproduce the worked case, identify an important failure condition, and independently solve the problems.

Concept and mechanism

A regulator drives a state toward zero; tracking a nonzero command requires feedforward, an error-integrator augmentation or another reference strategy. A prefilter can set the equilibrium gain for an accurate model, but uncertainty and disturbances can leave an offset. State penalties and effort penalties must be assessed alongside hard amplitude and slew constraints. Output tracking does not justify ignoring the actuator signal that causes it.

Governing relation: For x_dot=-a x+b u, u=-Kx+Nr; choose N=(a+bK)/b for unity constant-reference gain

To interpret this expression, mark each symbol as a state, command, observation, uncertainty, noise statistic or design weight. Determine whether the result is continuous-time, sampled-time, a static mapping, or a bound. The statement may require nonsingular matrices, bounded variation, a linear model or known timing. List the specific hypotheses before drawing an engineering conclusion.

Worked example

Take a=2 s^-1, b=1, K=3. The closed-loop plant becomes x_dot=−5x+Nr. Unity steady tracking requires N=5, giving x_ss=(5/5)r=r. If the actuator cap is ±2 and r=1, at initial x=0 the prefilter requests u=5, so the ideal predicted initial response is not feasible.

Independent check: Write the same example as an equation whose two sides must agree after substitution. Check the sign and limiting cases; then change only one number and predict whether the outcome increases, decreases, or requires a full recalculation. Do not mistake numerical agreement under synthetic data for physical validation. A result describing a covariance is conditional on the noise model; one describing stability is conditional on the model and its defined perturbations; one describing an optimum is conditional on both the objective and the feasible set.

Interpretation and decision

Prefilters work under the assumed static plant. Their lack of integral adaptation means that slow drift in a or b can create steady error. An augmented integral state changes system order and must be checked for controllability and closed-loop stability.

The design choice should identify the goal being improved—tracking, robustness, estimation, control effort or timing—and a quantity that may become worse. Distinguish a sufficient theorem condition from a necessary condition. For optimization explicitly report actuator limits; for estimation report which measurements support the inference. A calculated number is useful only when its assumptions and units are available to the reader.

When the argument fails

Setting the reference high enough can saturate an actuator, changing the feedback law from the Riccati design. LQR stability proofs for the ideal unconstrained linear loop cannot simply be applied after hard clipping.

Describe a falsifiable counterexample by naming an omitted dynamic, a violated constraint, an unobservable mode or a mischaracterized uncertainty set. Indicate which measured variable or computed check would reveal the discrepancy. Repeating the original calculation at the same nominal point cannot test a modeling assumption that never appeared in the equation.

Engineering case — Actuated environmental chamber

A model predicts temperature from heater power and measured wall temperature. A quadratic tracking penalty must be balanced against heater power and its rate of change, but power and temperature possess different units. Scale the weights and report a real input constraint. If the heater reaches its ceiling during warm-up, the unconstrained optimal controller is no longer the implemented law. An experiment can compare energy use, settling and temperature overshoot on the same synthetic disturbance trace.

Lab protocol: Cost accounting. Construct a table with physical units of each state and input; then show how the selected Q and R weights make their quadratic contributions comparable. Repeat at a second scale or command magnitude. If the optimum moves after simply changing reporting units, the weight transformation was inconsistent.

Create an assumption ledger listing sensor availability, actuator limits, initial state, operating point and the variable intentionally changed. For the main claim, separate what follows algebraically from what would require a second independent experiment. If the proposed experiment is unable to distinguish competing hypotheses, redesign it before collecting more repeated data.

Exercises

1. Calculation: For a=3, b=2 and K=1, calculate the prefilter N. For r=2, x0=0 and |u|≤3, determine whether the initial input is feasible. Show the intermediate arithmetic, a dimensions or sign check, and the condition under which the result is valid.

2. Alternative model: Construct an alternative interpretation of the same nominal observation, then name one additional signal that would separate the alternatives.

3. Sensitivity: Change one uncertain parameter by 20%. State which original claim remains logically valid, which result needs recalculation, and what new numerical input is required.

4. Connection: Compare the present method to Finite-horizon quadratic objectives. Name an assumption they share and one kind of answer each method cannot produce alone.

5. Verification: Draft a five-line test specification listing input, expected outcome, numeric tolerance, potential failure and the reason a passing simulation would not establish hardware safety.

Reference guidance

A complete answer to the calculation question begins from the displayed governing relation and shows substitutions. The alternative-model answer must predict a different outcome in a specified observation. The sensitivity answer must distinguish parameter error from measurement noise. The connection question should identify a difference in mathematical object or guarantee, not just a difference in terminology. A suitable test reports reproducible settings and a clear fail condition. Use the accompanying unit assessment for explicit numerical checks and worked answer guidance.

Unit 2 applied project — LQR cost-and-actuator tradeoff study

Engineering question

Use scalar xdot=−x+u with Q={0.25,1,4,9}, R=1. Compute the stabilizing Riccati solution, closed-loop pole and initial control for x0=2. Integrate a cost over a fixed interval with a stated numerical method. Impose |u|≤0.5 in an additional experiment and explain why the clipped law no longer inherits the original unconstrained optimality claim.

This is an offline design and evidence exercise. Its output is a reproducible mathematical study and a bounded conclusion, not permission to commission a physical controller. Write the study question in a sentence that could be falsified by a measured trajectory or a computed counterexample. Explain why the chosen input variation contains more information than an arbitrary random curve.

Setup, model and data protocol

Specify each variable, unit and initial state. The baseline model must be documented independently of the controller. List any linearization point, noise covariance, uncertainty interval, constraints or bounded disturbance. If the project contains a numerical experiment, provide either a short script or a CSV data table, and log the equation and numerical integration interval that produced it. Avoid a test in which the same output is used both to choose every model parameter and to certify accuracy. If this project cannot be evaluated without a missing sensor or unsupported assumption, make the gap an explicit result.

Required analytical work

1. Derive the governing state equation, transfer equation, cost, covariance update or stability inequality appropriate to this unit. Where possible, independently derive a special case without a library solver.
2. Calculate a baseline numerical value and check it using a distinct method: direct substitution, a matrix identity, a simulated recurrence or an analytically known limit. Include the expected dimensions.
3. Declare a testable range for one key parameter. Sample several values and describe what is and is not proven by that sample. For a mathematical worst-case guarantee, state why the chosen analytic bound applies to the full nominated uncertainty set.
4. Check performance and feasibility separately. A low cost does not imply constraints are met; a stable trajectory does not prove that an actuator command is allowed. Where the optimizer or model predicts an impossible demand, report infeasibility plainly.
5. Record at least one deliberately adverse scenario: sensor bias, delayed packet, perturbation outside the fit range, insufficient excitation, a saturated actuator, or a change in the model’s region of validity.

Measurement and critical interpretation

Cost accounting: Construct a table with physical units of each state and input; then show how the selected Q and R weights make their quadratic contributions comparable. Repeat at a second scale or command magnitude. If the optimum moves after simply changing reporting units, the weight transformation was inconsistent.

A technically useful report separates mathematical truths under explicit assumptions, simulations of those assumptions, and empirical claims that require physical evidence. Explain which additional variable would most improve confidence. A compelling graph should show labeled axes and a model-versus-reference comparison, not just a beautiful output trace. If a proposed controller improves response time but worsens effort, the tradeoff belongs in the conclusion. Do not discard a failed scenario merely because it disrupts the nominal result.

Required report package

• Model card: signals and dimensions, operating range, nominal and uncertain parameters, initial state and constraints.
• One derivation checked by a second independent calculation, with all rounding disclosed.
• A baseline numeric table and an adverse-case numeric table, each with reproducible values.
• At least two plots with captions explaining what the curves demonstrate and what they do not.
• A decision paragraph about the next cheapest test that could reject an important assumption.
• A separate limitations note identifying implementation, sensor, timing, actuator and safety evidence not supplied by this exercise.

Assessment rubric

25% analytical correctness means the mathematical object and its conditions are identified precisely. 20% numerical traceability means the results can be recomputed from the stated data. 20% comparison requires a meaningful alternative, sensitivity or failure scenario. 20% engineering interpretation requires an explicit distinction between a passed model test and an unverified deployment assumption. 15% communication covers units, graph labels, repeatability and clear identification of the limits of conclusions. Strong work may conclude that a proposed design is not justified by the evidence; that is not a failure of the assignment.

Unit 2 assessment — Optimal control and quadratic design

Instructions: Show the derivation and dimensions, not only the final value. Work independently before reading the reference key. Every number refers to a synthetic model and should not be interpreted as a device setting. The assessment combines computation, experimental design, and evaluation of whether a claim follows from the evidence.

Part A — quantitative short questions

A1. Given x0=2, Qf=4,R=1 in one-step integrator, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A2. Given Riccati scalar A=B=Q=R=Pterminal=1, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A3. Given A=−1,B=1,Q=R=1 continuous LQR, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

A4. Given a=2,b=1,K=3, unity prefilter, calculate the named mathematical quantity. State the model assumptions that make the requested result well defined. Explain whether the answer represents a transfer characteristic, a state estimate, an optimal action, a bounded inequality or a verification verdict. Show the intermediate arithmetic and one reason an actual data set might disagree.

Part B — distinct lesson problems

B1. Finite-horizon quadratic objectives. Repeat the one-step problem for R=4 and Qf=4. Compute the optimum and interpret how the increased input penalty changes the residual state. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B2. Bellman recursion and the Riccati step. Starting with P2=2 and Q=R=A=B=1, compute P1 and K1. Explain why the gain changes relative to the example. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B3. Infinite-horizon LQR and a scalar analytic result. Solve the same scalar problem for Q=4 and R=1. Select the stabilizing Riccati root and calculate the new closed-loop pole. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

B4. Reference tracking, prefilters and control constraints. For a=3, b=2 and K=1, calculate the prefilter N. For r=2, x0=0 and |u|≤3, determine whether the initial input is feasible. Include the model, numerical or logical solution, and a limiting-case check. Then design one follow-up measurement or computed counterexample that can challenge the result under an untested assumption.

Part C — synthesis and model criticism

C1. Data and claims. A team plots attractive curves on its calibration data and calls the controller robust and safe. Identify the exact robust or feasible property that would need to be established, a missing uncertainty or fault class, and a test that could falsify the unsupported conclusion. Avoid generic assurances.

C2. Evidence hierarchy. Separate a theoretical identity, a result from a discrete simulation, a laboratory observation and a physical safety approval. For the current unit, provide one concrete example of each, then explain why one cannot silently replace another.

C3. Design comparison. Select two approaches studied in this unit. Compare their required model detail, computational or measurement burden, failure sensitivity and interpretability. Name one operating condition in which the less elaborate approach could be preferable.

C4. Integration with Book 11. Start from a textbook PID or linear state-space example and show which assumption must be changed to require the advanced tool from the present unit. Identify an actuator constraint, sensor characteristic or nonlinear effect the simple representation had omitted.

Reference answers and grading notes

A1 reference. x0=2, Qf=4,R=1 in one-step integrator leads to u*=−1.6, x1=0.4 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A2 reference. Riccati scalar A=B=Q=R=Pterminal=1 leads to K=0.5 and Pprevious=1.5 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A3 reference. A=−1,B=1,Q=R=1 continuous LQR leads to P=√2−1 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

A4 reference. a=2,b=1,K=3, unity prefilter leads to N=5 in the stated ideal model. The answer is incomplete without the numerical relationship, relevant units, and an explicit condition such as positivity, stability, nonsingularity, observability or a known sample interval. Where an inequality appears, the conclusion is restricted to that bound and the defined uncertainty class; where an estimated value appears, the inference is conditioned on its assumed noise model.

B series marking. Award credit for four independent elements: correct mathematical relationship, transparent substitution, meaningful limiting-case or sensitivity check, and a proposed measurement that would discriminate among possible models. The four named lesson scenarios demand different reasoning; a generic statement about noise or risk is not sufficient. Problems involving constraints must state whether a proposed solution is feasible rather than only desirable.

C series marking. Accept conclusions that identify an information gap rather than inventing missing values. A strong answer describes the exact signal at which the model would first fail, a safe way to expose that failure in software, and which human or independent engineering review would be needed for any later physical test. For stochastic examples insist on a distinction between model variance and guaranteed error. For optimization distinguish an unconstrained minimum from a feasible command. For robust control do not confuse sufficient tests with necessary ones.

Post-assessment research note

Write a final one-page technical summary with the columns: claim, equation, required assumptions, numerical evidence, strongest counterexample and next verification step. The purpose is to practice deciding what is established, what is conditional and what remains unknown. A model may be educationally useful even when publication-quality validation is absent. Never treat an optimized cost, a stable nominal trace, or an intact software archive as an independent functional-safety assessment.